6 research outputs found
On strongly chordal graphs that are not leaf powers
A common task in phylogenetics is to find an evolutionary tree representing
proximity relationships between species. This motivates the notion of leaf
powers: a graph G = (V, E) is a leaf power if there exist a tree T on leafset V
and a threshold k such that uv is an edge if and only if the distance between u
and v in T is at most k. Characterizing leaf powers is a challenging open
problem, along with determining the complexity of their recognition. This is in
part due to the fact that few graphs are known to not be leaf powers, as such
graphs are difficult to construct. Recently, Nevries and Rosenke asked if leaf
powers could be characterized by strong chordality and a finite set of
forbidden subgraphs.
In this paper, we provide a negative answer to this question, by exhibiting
an infinite family \G of (minimal) strongly chordal graphs that are not leaf
powers. During the process, we establish a connection between leaf powers,
alternating cycles and quartet compatibility. We also show that deciding if a
chordal graph is \G-free is NP-complete, which may provide insight on the
complexity of the leaf power recognition problem
Local Certification of Some Geometric Intersection Graph Classes
In the context of distributed certification, the recognition of graph classes
has started to be intensively studied. For instance, different results related
to the recognition of planar, bounded tree-width and -minor free graphs have
been recently obtained. The goal of the present work is to design compact
certificates for the local recognition of relevant geometric intersection graph
classes, namely interval, chordal, circular arc, trapezoid and permutation.
More precisely, we give proof labeling schemes recognizing each of these
classes with logarithmic-sized certificates. We also provide tight logarithmic
lower bounds on the size of the certificates on the proof labeling schemes for
the recognition of any of the aforementioned geometric intersection graph
classes
Polynomial kernels for edge modification problems towards block and strictly chordal graphs
We consider edge modification problems towards block and strictly chordal
graphs, where one is given an undirected graph and an integer and seeks to edit (add or delete) at most edges from to
obtain a block graph or a strictly chordal graph. The completion and deletion
variants of these problems are defined similarly by only allowing edge
additions for the former and only edge deletions for the latter. Block graphs
are a well-studied class of graphs and admit several characterizations, e.g.
they are diamond-free chordal graphs. Strictly chordal graphs, also referred to
as block duplicate graphs, are a natural generalization of block graphs where
one can add true twins of cut-vertices. Strictly chordal graphs are exactly
dart and gem-free chordal graphs. We prove the NP-completeness for most
variants of these problems and provide vertex-kernels for Block Graph
Edition and Block Graph Deletion, vertex-kernels for Strictly Chordal
Completion and Strictly Chordal Deletion and a vertex-kernel for
Strictly Chordal Edition